Research ArticleGliding-Guided Projectile Attitude Tracking Controller DesignBased on Improved Adaptive Twisting Sliding Mode Algorithm

Wenguang Zhang and Wenjun Yi

National Key Laboratory Transient Physics, Nanjing University of Science and Technology, Nanjing 210094, China

Correspondence should be addressed to Wenjun Yi; njustwz@163.com

Received 30 January 2020; Revised 18 September 2020; Accepted 1 October 2020; Published 31 October 2020

Academic Editor: Antonio Concilio

Copyright © 2020 Wenguang Zhang and Wenjun Yi. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

The finite-time attitude tracking control for gliding-guided projectile with unmatched and matched disturbance is investigated. Anadaptive variable observer is used to provide estimation for the unmeasured state which contains unmatched disturbance. Then, animproved adaptive twisting sliding mode algorithm is proposed to compensate for the matched disturbance dynamically with bettertransient quality. Finally, a proof of the finite-time convergence of the closed-loop system under the disturbance observer and theadaptive twisting sliding mode-based controller is derived using the Lyapunov technique. This attitude tracking control schemedoes not require any information on the bounds of uncertainties. Simulation results demonstrate that the proposed methodwhich is able to acquire the minimum possible values of the control gains guaranteeing the finite-time convergence performswell in chattering attenuation and tracking precision.

1. Introduction

The most significant difference between gliding-guidedprojectile (GGP) and traditional projectile is that the for-mer one is equipped with wing assembly. On the onehand, the wing assembly makes the projectile be capableof gliding, which enables the gliding-guided projectile tostrike the target from a greater distance. On the otherhand, the wing assembly provides the projectile with theability of controlling its flight path, which means that thegliding-guided projectile can achieve precise strikes [1, 2].Due to these advantages, the gliding-guided projectile hasbecome the current research hotspot. It should be notedthat the main way to ensure the stability of a projectile’sairframe is a high body roll-axis spin rate, which is a sig-nificant difference from traditional missiles, and is thesource of nonlinear coupling between the yaw and pitchchannel of GGP. Therefore, the design of GGP, especiallythe design of the attitude controller, is a challenging task.This is due to the fact that the GGP model suffers from

high nonlinearity and strong coupling characteristics. Inaddition, a GGP is vulnerable to various disturbances dur-ing flight, such as the aerodynamic parameter perturba-tion, the unmodeled dynamics, and strong and suddenwind gusts which change greatly along the latitude andlongitude [3].

To compensate for uncertainties and disturbances suf-fered by the flight control system, a lot of works have beencarried out. In [4], the authors propose a pitch attitudeand lateral decoupling control based on active disturbancerejection control for aircraft. To obtain robustness, theyutilize an extended state observer (ESO). In [5], thefinite-time attitude control for a reusable launch vehiclewith unmatched disturbance is investigated, and they pro-pose an adaptive multivariable disturbance compensationmethod to estimate the disturbance. In [6], the adaptivecontrol on the longitudinal dynamics of a hypersonic flightvehicle in the presence of wind effects is investigated. Theunknown disturbance due to wind is estimated by neuralnetworks. In [7], they propose a dual closed-loop control

HindawiInternational Journal of Aerospace EngineeringVolume 2020, Article ID 1480427, 15 pageshttps://doi.org/10.1155/2020/1480427

https://orcid.org/0000-0002-1665-9622https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/1480427

framework for attitude control of a quadrotor. In the dual-loop framework they proposed, active disturbance rejec-tion control and proportional-derivative control are usedin the inner and outer loops, respectively. To cancel theeffect of dynamic disturbance, they use an ESO. In [8],they utilize the prescribed performance control techniqueto tackle the issue of attitude control of hypersonic flightvehicles (HFVs), which largely improves the transientcharacteristics of HFVs. In [9], the authors propose anintelligent flight control scheme based on reinforcementlearning technique. Besides, many other control methods,such as fuzzy control, neural network control, backstep-ping control, and model following control, have beenapplied on this issue [10–20]. It should be noted that var-ious feedback linearizations have been applied to this issue[21–23]. The aims of such methods are to cancel the non-linear dynamics via mathematical transformation so as tocontrol the system in a linear manner. It is able to providea full envelope nonlinear flight control system. However,the design techniques need much information about thesystem; otherwise, the output will not be desired.

Higher-order sliding mode (HOSM) control is anotherpopular method to deal with an uncertain system. A twist-ing controller (TC) is historically the first 2-SMC algo-rithm [24] that drives the output and derivative of thesystem to the origin in finite time in the presence of distur-bance whose boundary is known. If the boundary of the dis-turbance is unknown, it is necessary to design adaptivetwisting sliding mode control (ATSMC) methods to satisfythe condition for sliding mode to exist. One efficient schemeis designing algorithms which are able to adjust dynamicallythe control gains. In [25], this kind of adaptive twisting slid-ing mode control is integrated into attitude tracking control-ler for the reentry reusable launch vehicle. In [26], thetwisting sliding mode control law is modified with a pro-posed gain adaptation strategy to improve the attitude track-ing performance of quadrotor unmanned aerial vehicles. In[27], the Lyapunov theory is used to design the ATSMC.However, the aforementioned methods fail to search theminimum possible value of the control gain. As we know,the amplitude of chattering which hinders the applicationsof the sliding mode control is proportional to the magnitudeof discontinuous control.

In [28], an adaptive algorithm is designed via equiva-lent control. Their method is able to search the minimumpossible value of control. In their work, the gains can beadjusted adaptively to compensate for the disturbanceswith lower values without knowledge of the bounds ofthe disturbances. However, this method may lead to largechattering when the values of the gains are close to theideal ones. This is due to the fact that the changing rateof the gains still keeps a large value in such circumstances.

Motivated by the aforementioned work, we propose arobust attitude tracking scheme for GGP with chatteringattenuation in this paper. The error dynamics of the atti-tude tracking system suffers from both matched andunmatched disturbances, where the bounds of disturbancesare not known. In order to estimate the unmeasured stateinfluenced by unmatched disturbance, an adaptive observer

is utilized. Then, an improved equivalent control-basedadaptive twisting sliding mode control (IECATSMC) algo-rithm for [28] is designed and adopted by the controller tocompensate for the matched disturbance dynamically. Themain contributions of this paper can be summarized asfollows:

(i) The improved adaption algorithm is able to adap-tively adjust the changing rates of the gains on alarger scale. Thus, the transient characteristics ofthe adaptation strategy in [28] are improved. Sincethe proposed control scheme is able to acquire theminimum possible values of the control gains, itperforms better in chattering attenuation than thefixed gain TSMC and the existing ATSMC, whichcan be seen from the simulation results given inthe latter

(ii) A high-performance disturbance estimator pro-posed in [29] is used in this paper to reconstructthe unmeasurable system state, which extends theapplication of this estimator. More importantly, bydoing so, the finite-time convergence property ofthe closed-loop system is guaranteed

(iii) A proof of the finite-time convergence of the closed-loop system under the observer and theIECATSMC-based controller is derived using theLyapunov stability theory

The paper is organized as follows: the GGP attitudekinematics model is given in Section 2; then, error dynam-ics of attitude tracking system is derived. To estimate theunmeasured state in error dynamics, an adaptive observeris utilized in Section 3. The improved adaptive twistingalgorithm which is able to search the minimum possiblevalues of the control gains is investigated for the attitudecontrol in Section 4, and a proof of the finite-time conver-gence for the overall system is derived. The simulationresults are provided in Section 5. Finally, concludingremarks are summarized in Section 6.

Notations. Throughout the paper the following notationswill be used. For a given vector x = ½x1,⋯, xi,⋯, xn�T ∈R,denote ∥x∥ =

ffiffiffiffiffiffiffiffixTx

p. For a given scalar variable y, denote ∣y ∣

as the absolute value of y.

2. Gliding-Guided Projectile AttitudeKinematics Model

In actual flight, the speed of the GGP is variable due to thevarying thrust and drag. However, if we study the dynamicsof the projectile which is gliding along a very short flightpath, we can safely think that the flight speed and rotationalspeed of the projectile are constant. Besides, we can alsoneglect the variation of the dynamic pressure and Machnumber as well as other physical parameters. Then, underthe assumption that the attack angle and side slip angle aresmall, the attitude kinematics model of the gliding-guidedprojectile can be constructed as [30, 31]

2 International Journal of Aerospace Engineering

_θ = α1α − α2β,− _φ cos θ = α1β − α2α,€ϑ = b1α − b2β + b4δz − α3δy − b3 _ϑ + b5 cos ϑ:ψ − b5 _ϑ,

€ψ cos ϑ = b1β + b2α + b4δy − α3δz − b3 cos ϑ: _ψ − b5 _ϑ,

8>>>>>>>>>>>:ð1Þ

where ai and bi denote kinetic coefficients, where a1 =ðQS/mvÞðCαL + CδLÞ, a2 = ðQS/mvÞCμ′′ð _γd/vÞ, a3 = ðQS/mvÞCδL, b1 = ðQS/EÞðmz′ +mσ′Þ, b2 = ððQSlγ̇DÞ/EvÞmy′′, b3 = ðQSlD/EvÞmzz′ , b4 = ðQSl/EÞmσ′, and b5 = C _γ/E.

By accepting that the attitude change speed is more fasterthan the change speed of the velocity vector direction of theprojectile center of mass on the short flight path [30], andwithout loss of generity, we assume that _ϑ ≈ _α, _φ ≈ _β. From(2) and the assumption of α = ϑ − θ, β = ðψ − φÞ cos θ, andcos θ ≈ cos ϑ, we obtain

_α = _ϑ − a1α + a2β,_β = _ψ − _φð Þ cos ϑ + ψ − φð Þ _ϑ sin ϑ,

d ϑ − θð Þdt2

= −k1 _α − k2 _β + k3α − k4β +

k5δz − k6δy,d ψ − φð Þ cos θð Þ

dt2= −k1 _β − k2 _α + k3β + k4α

+k5δy − k6δz ,

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð2Þ

where k1 = b3 + a1, k2 = b5 + a2, k3 = b1 + b3a1 + b5a2, k4 =b2 − b3a2 − b5a1, k5 = b4, and k6 = a3.

Assume that the steering gear system of the projectile isdesigned as a second-order one. For the convenience ofstudy, we only consider the steady output of equivalent rud-der deflection [32]:

δz

δy

" #= kskr

cosγd sin γd−sin γd cos γd

" #δzc

δyc

" #, ð3Þ

where

ks =1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 − T2s _γ2� �2 + 2μsTs _γð Þ2q ,

kr = τd _γ + arccos1 − T2s _γ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 − T2s _γ2� �2 + 2μsTs _γð Þ2q ,

8>>>>>>>>>>>:ð4Þ

where δyc and δzc stand for equivalent rudder deflect anglecontrol commands for pitching channel and yaw channel,respectively; ks is the gain of canard; Ts denotes the time con-stant of steer gear system; μs denotes the system dampingratio; τd represents delay time;, and γd represents delay phaseangle.

Taking matched and unmatched disturbances into con-sideration, from ((1)) and ((2)), we get

_α = _ϑ − a1α + a2β + d1,_β = _ψ − _φð Þ cos ϑ + ψ − φð Þ _ϑ sin ϑ + d2d ϑ − θð Þdt2

= f1 + Δf 1 + A1ΘT +

B1 + ΔB1� �

U + _d1,d ψ − φð Þ cos θð Þ

dt2= f2 + Δf2 + A2Θ

T + B2 + ΔB2� �

U + _d2,

8>>>>>>>>>>>>>>>>>>>>>>>:ð5Þ

where

Θ = α β½ �T , ð6Þ

U = δyc, δzc� �T , ð7Þ

f1 = −k1 _α − k2 _β + k3α − k4β, ð8Þ

f2 = −k1 _β − k2 _α + k3β + k4α, ð9Þ

A1 = k3  − k4½ �T , ð10Þ

A2 = k4 k3½ �T , ð11Þ

B1 = κ6 κ5½ �T , ð12Þ

B2 = κ5  − κ6½ �T , ð13Þκ5 = k5kskr cos γd − k6kskr sin γd , ð14Þκ6 = k5kskr sin γd + k6kskr cos γd , ð15Þ

besides, Δf1 , Δf2 , ΔB1 , and ΔB2 are uncertainties induced byparameter variation and model simplification; d1, d2 standfor external unmeasurable disturbances.

The object is to design a control inputU = ½δyc, δzc�T so asto drive the attitude angle vectorΘ to follow the control com-mand Θc = ½αc, βc�T in spite of the internal and externaldisturbances.

Thus, defining the attitude tracking error as eΘ =Θ −Θc,using (1) and recalling the relationships α = ϑ − θ and β =ðψ − φÞ cos θ, the error dynamics for (5) can be calculatedas

_eΘ = zΘ,_zΘ = F+AΘ + BU − €Θc + Δ,

(ð16Þ

3International Journal of Aerospace Engineering

where

zΘ =_ϑ − a1α + a2β

_ψ − _φð Þ cos ϑ + ψ − φð Þ _ϑ sin ϑ

" #− _Θc + Δ1, ð17Þ

Δ1 = d1 d2½ �T , ð18Þ

F= f1 f2½ �T ð19Þ

A = A1 A2½ �T , ð20Þ

B = B1 B2½ �T , ð21Þ

Δ =Δf1 + ΔU1Δf2 + ΔU2

" #+ _Δ1, ð22Þ

where the deviations ΔU1 = ΔB1U, ΔU2 = ΔB2U induced bythe uncertainties in B1 and B2 can be seen as internaldisturbances.

The following assumption adopted from [29] is needed tothe existences of causal control schemes for system (5).

Assumption 1. The disturbances Δf1 , Δf2 , ΔB1 , ΔB2 , d1, and d2and the first-order derivatives of them are bounded, but theboundaries are unknown.

3. Finite-Time Adaptive Observer for zΘAs we see from (6), the error state contains unknown distur-bance Δ1; therefore, it cannot be measured directly. Toaddresss this problem, an adaptive finite-time variableobserver is used for (5) in this section.

Assumption 2. Suppose that the disturbance ∥Δ∥≤C and∥ _Δ∥≤C′, where C and C′ are unknown.

For system (5), the controller based on twisting algorithmcan be designed as

U = 1B −F−AΘ −€Θc − r1

eΘ∥eΘ∥

− r2zΘ∥zΘ∥

� �, ð23Þ

where r1, r2 > 0 are control gains. It should be noted that thecontoller (16) is impractical because of the existence oflumped disturbance Δ, whose upper bound is unknown.Meanwhile, as mentioned before, the error state zΘ is notmeasurable, so an adaptive finite-time observer is needed.Inspired by [29, 33], an adaptive finite-time observer for(16) is formulated as

_ϖ1 = −kΓ1sΓ

∥sΓ∥1/2+ ϖ2,

_ϖ2 = −kΓ2sΓ∥sΓ∥

, ð24Þ

where the auxiliary sliding variable

sΓ = ϖ1 − eΘ, ð25Þ

and the adaptive gains kΓ1 and kΓ2 are

_kΓ1 =ω1

ffiffiffiι12

r, if∥sΓ∥>σT ,

0, if∥sΓ∥≤σT ,

8>:kΓ2 = ε1kΓ1 ,

8>>>>>>>:ð26Þ

where σT represents the arbitrary small threshold value, andω1, ι1, and ε1 are positive constants. It can be concluded thatzΘ can be estimated by the term ϖ2 in finite time regardless ofthe unknown bounded uncertainty Δ [29].

4. Main Results

If r1 and r2 in (16) satisfy r1 > 2C and r2 > C, the systemrepresented by (5) is stable [34]. However, to make tradi-tional twisting sliding mode control strategies meet thedemand is a challenging thing because the upper boundof disturbance is often unknown especially in the case ofgliding-guided projectile. Therefore, adaptive methodswhich can adjust the values of control gains have beenproposed [27]. It should be noted that such methods usu-ally cause large gains, which may result in an undesirableaggressive chattering phenomenon. In this section, animproved adaptive twisting sliding mode control methodis developed to search the minimum possible values ofcontrol with better transient quality.

Using ẑΘ estimated in (17), the actual adaptive twistingcontroller is then designed as

U = 1B −F−AΘ −€Θc − r1

eΘ∥eΘ∥

− r2ẑΘ∥ẑΘ∥

� �, ð27Þ

where the estimation ẑΘ = ϖ2 = zΘ + e with e represents theestimation error of zΘ. Substituting (25) into (5), theclosed-loop system is changed into

_eΘ = ẑΘ − e,

_zΘ = −r1eΘ∥eΘ∥

− r2ẑΘ∥ẑΘ∥

+ bΔ ,8>: ð28Þ

where bΔ = _e + Δ.

4 International Journal of Aerospace Engineering

4.1. Improved Adaptive Twisting Sliding Mode ControlAlgorithm Design. To simplify the adaptive controller design,the ideal estimate of zΘ will be exploited, and the followingrepresentation for (26) can be considered:

_x1 = x2,

_x2 = −h1 tð Þx1∥x1∥

− h2 tð Þx2∥x2∥

+ f,

8 2D0, then the states x1 and x2 will converge to ori-gin in finite time.

The proof of Lemma 3 is given in Appendix A.

Thus, the problem to be solved in the following partbecomes to design a gain adaptive algorithm, which ensuresthe condition L > 2D0 holds without any knowledge aboutthe disturbance f .

Theorem 4. For the uncertain systems (27), the dynamics ofthe control gains are designed as h1 = LðtÞ, h2 = 0:5LðtÞ, whereLðtÞ is defined as

_L = eρ ξj j−ζð ÞηL sign ξð Þ −M L − L+½ �+ +M μ − L½ �+,

z½ �+ ≔1 if z ≥ 0,0 if z < 0,

( ð30Þ

where ρ, ζ, η, M, and L+ are positive constants, and they sat-isfy M > eρð∣ξ∣−ζÞηL+, L >Dp0. ξ satisfies

ξ = ∥ x1∥x1∥

+ 0:5 x2∥x2∥

eq

∥−ε, ð31Þ

where ½−�eq is called the equivalent control [23] function,which is an average value ranging from (-1, 1). ε is a pos-itive constant and satisfies ε ∈ ð0; 0:5Þ. Then, the states xiði = 1, 2Þ will converge to origin in finite time.

Proof. The analysis can be divided into three phases.

Phase 1. the gain L satisfies L < 2D0.

The system (28) is on reaching phase, so we have

∥x1∥x1∥

− 0:5 x2∥x2∥

eq∥ >∥ x1

∥x1∥∥−0:5∥ x2

∥x2∥∥ = 0:5:

ð32Þ

Recall that ε ∈ ð0,0:5Þ, thus

ξ = 0:5 − ε > 0, ð33Þ

so the value of L will keep increasing until it is large enough,which means

Ð t1t0eρð∣ξ∣−ζÞηL sign ðξÞ −M½L − L+�+ +M

½μ − L�+dt > 2D0 can be achieved.Assuming L < l+, we can have

_L ≥ ηL +M μ − L½ �+ ≥ ηL, ð34Þ

here, eρð∣ξ∣−ζÞ ≥ 1 is used. Therefore, by solving the inequationL ≤ 2D0, we can obtain t1 ≥ ln ð∣2D0 − Lðt0Þ + eηt0 ∣ Þ, whichmeans the condition L ≤ 2D0 can be achieved at most afterln ð∣2D0 − Lðt0Þ + eηt0 ∣ Þ.

Phase 2. the gain L satisfies L > 2D0.

According to Lemma 3, the states x1 and x2 will con-verge to the origin in finite time. Furthermore, the con-

vergence time t2 satisfies t2 ≤ t0 + t1 + ð16�λ3/4maxðPÞV1/40ðt1ÞÞ/�E, where the definitions of P and �E can be seenin Appendix A.

Phase 3. the system enters the sliding mode.

First, Let us assume that L ∈ ½μ, L+�, which means ∥f∥/ε> μ, and the time derivative of ½sign ðx1Þ + 0:5 sign ðx2Þ�eqexist. A Lyapunov function candidate is defined as VðξÞ =ð1/2Þξ2. Since system (27) has entered the sliding mode inphase 3, the following equation holds

∥ueq∥ = L∥x1∥x1∥

+ 0:5 x2∥x2∥

eq∥ = ∥f∥, ð35Þ

where ueq = L½ðx1/∥x1∥Þ + 0:5ðx2/∥x2∥Þ�eq is defined asequivalent control input. Taking the time derivative of

5International Journal of Aerospace Engineering

VðξÞ, assuming that LðtÞ ∈ ½μ, L+�, which means that ∥f∥/ε > μ and using (28), (29), and (33) yield

_V0 ξð Þ = _ξξ= ξddt

∥x1∥x1∥

+ 0:5 x2∥x2∥

eq

����� = ξ ddt ∥f∥L� �

= −∥f∥ξL−2 _L + ξL−1 ddt

∥f∥ð Þ= −∥f∥ξL−2 eρ ∣ξ∣−ζð ÞηL sign ξð Þ −M − L − L+½ �+

� +M μ − L½ �+

�+ ξL−1fT _f∥f∥−1

≤ −εeρ ∣ξ∣−ζð Þημ∣ξ∣L−1η∣ξ∣ + ∣ξ∣L−1D1= −∣ξ∣L−1 εeρ ∣ξ∣−ζð Þημ −D1

� ð36Þ

If η >D1/ðεμ ∣ eρð∣ξ∣−ζÞÞ, we will have

_V ξð Þ ≤ −ffiffiffi2

p εeρ ξj j−ζð Þημ −D1�

L+

ffiffiffiffiffiffiffiffiffiffiV ξð Þ

q: ð37Þ

Therefore, VðξÞ = 0 is achievable, at least after t f = L+/ðεeρð∣ξ∣−ζÞημ −D1Þ ∣ ξðt2Þ ∣ . After the adaption process forL is over ðt > t f Þ, we obtain

∥x1∥x1∥

+ 0:5 x2∥x2∥

eq∥ = ∥f∥

L= ε, ð38Þ

which means L = ∥f∥/ε. Another case is that ð∥f∥Þ/ε > μ,then L increases until LðtÞ = μ, and it will be maintainedat that level. The theorem is proven.

If ε is very close to 0:5, the minimal possible gain will befound for the current value of disturbance. As a result, theamplitude of chattering will be reduced.

Remark 5. The main feature of the improved adaptive algo-rithm is that it can adaptively adjust the changing rate ofthe gain L, which is realized by a regulatory factor eρð∣ξ∣−ζÞ.The factor enables the changing rate of L to be larger orsmaller than η. In general, when ∣ξ ∣ becomes larger than ζ,which means L is far away from its ideal value, eρð∣ξ∣−ζÞ canalso increase and become larger than 1. Hence, L will tendmore quickly toward its ideal value. On the other hand, once∣ξ ∣ enters a small region Cξ = f∣ξ∣≤ζg, which means L hasbeen already very close to its ideal value, eρð∣ξ∣−ζÞ will becomesmaller than 1. As a result, the amplitude of gain oscillationwill be reduced so that the condition L > 2∥Δ∥ can be safelyensured.

Remark 6. The function ½ðx1/∥x1∥Þ + 0:5ðx2/∥x2∥Þ�eq can bederived by low pass filter

τ _z1 + z1 =x1∥x1∥

+ 0:5 x2∥x2∥

, z1 = 0, 0½ �T , ð39Þ

with a small time constant τ > 0, respectively. The output z1satisfies

∥z1 −x1∥x1∥

+ 0:5 x2∥x2∥

eq

∥ ≤O τð Þ��!τ→0

0: ð40Þ

Attitude kinematicmodel

Eqs. (2-6)

�e tracking error dynamicsEqs. (7, 8)

Gliding guided projectile attitude kinematic model

IECATSMCAdaptive observer

ControllerEq. (13)

Adaptive gainsEqs. (16, 17) Eqs. (10-12)

Figure 1: Schematic diagram of the proposed control strategy.

Table 1: Parameters for a gliding-guided projectile.

b1/s−2 b2/s−2 b3/s−2 b4/s−2 b5/s−2

−129:4 0 0:291 90:519 1:886a1/s−1 a2/s−1 a3/s−1 _γ r/sð Þ τ/s0:287 0 0:136 10 0:015ks Ts/s μs δmax/o _δmax/ rad/sð Þ1 1/150 0:7 20 200

6 International Journal of Aerospace Engineering

4.2. Stability Analysis for the Closed-Loop System. In this sub-section, the stability of the gliding guided projectile closed-loop system (5) is analyzed.

Theorem 7. For the attitude system (5), the adaptive twistingcontroller is designed as (25), with the adaptive gains definedas r1 = LðtÞ, r2 = 0:5LðtÞ, where LðtÞ is obtained from Theo-rem 4; then, the states eΘ, zΘ will converge to the origin infinite time.

The proof of Theorem 7 is given in Appendix B.

In order to illustrate the entire control architecture in thispaper, the schematic diagram is shown in Figure 1.

5. Simulation and Results

To verify the effectiveness of the adaptive algorithm-basedcontroller, the attitude motion of a certain type of GGP

Table 2: Control parameters of the three algorithms.

Method Parameters

IECATSMC ρ = 2, ζ = 0:07, η = 5:59, L+ = 50, μ = 0:1, M = 279:5279, ε = 0:49, ω1 = 4:9, ι1 = 2:4, ε1 = 0:6, σT = 10−3

ATSMC [34] Lmin = 32, ϖ1 = 4:9, γ1 = 1:1, μ1 = 0:0004, χ = 40CTSMC [24] L = 20

Estim

atio

n er

ror o

f z𝜃

(deg

)

0 5

0

0.5

–0.5

1

–110

Time (sec)15 20

(a)

00

1

2

3

𝜅Γ

𝜅Γ1

𝜅Γ2

4

5 10Time (sec)

15 20

(b)

Figure 2: The curves of estimation errors and adaptive gains of the observer.

0 2

ECATSMC

ATSMC

CTSMC

IECATSMC

4

𝛼c

6 8 10 12 14 16 18Time (sec)

4

6

8

10

12

14

16

𝛼 (d

eg)

0.3 0.35 0.4 0.45 0.5 0.55 0.6

10.8

10.9

11.9 11.95 1210.98

11

(a)

–10

–5

0

5

10

𝛽 (d

eg)

5.15 5.16 5.17–0.01

0

0.01

7.5 8 8.5 9–0.5

00.5

1

0.2 0.4 0.6 0.8 1 1.2

0

1

2

0 2

ECATSMC

ATSMC

CTSMC

IECATSMC

4

𝛽c

6 8 10 12 14 16 18Time (sec)

(b)

Figure 3: Tracking effect of yaw and pitch commands.

7International Journal of Aerospace Engineering

is taken as the simulation project. The parameters of theGGP are shown in Table 1. The initial values for the plantare xð0Þ = 0:16 0 −0:1047 0½ �T . Reference commandsare taken as αc = cos ð0:5πtÞ + 10 deg, βc = 0 deg.

To obtain the equivalent control information, we usea low-pass filter formulated as (37) in the simulation,and the parameter is set as τ = 0:015. Simulations areperformed in MATLAB/simulink environment, where

00

10

20

30

40

50

2

L – ECATSMC

L – ATSMC

L-a

dapt

ion

L – CTSMC

L – IECATSMC

4

2||△∣∣

6 8 10 12 14 16 18Time (sec)

⌃

Figure 4: Adaptive gain curves.

0 2 4 6 8 10 21 14 16 180

2

4

6

0.61970.81971.01971.21971.41971.5664

0.6 1 1.4 1.8 9 10 11 120.80.9

11.11.2

Value=1

𝑒𝜌(|ξ

|—ζ)

(a)

0 0.5 1 1.5Time (sec)

0

5

10

15

20

25

10 11 12Time (sec)

10

15

20

L – ECATSMC

L – IECTASMC

2||△∣∣⌃

(b)

Figure 5: Comparison between IECATSMC and ECATSMC.

8 International Journal of Aerospace Engineering

sample time is set as 0.001 s, and fourth-order Runge-Kutta method is chosen as the numerical integrationmethod.

5.1. Comparison. In the simulation, severe disturbances areconsidered. Δf1, Δf2 are the 40% uncertainty in f1, f2, respec-tively. ΔB1 and ΔB2 are the 10% uncertainty in B1 and B2,respectively. The external disturbances are

d1 =9t − 8 cos t2

� �− 14 cos t5

� �, 0 ≤ t ≤ 6,

4 9t − 8 cos t2

� �− 14 cos t5

� �� �, t > 6,

8>>>>>:

d2 =t + 2

πcos πt2

� �−

3πcos πt3

� �, 0 ≤ t ≤ 6,

4 t + 2π

cos πt2

� �−

3π

cos πt3

� �� �, t > 6:

8>>>>>: ð41Þ

For performance comparison, the ATSMC proposed in[34] and CTSMC proposed in [24] are also simulated underthe same conditions. The gain adaptive schemes are formu-lated as

(i) ATSMC [34]: the gain adaptive scheme _L is definedas

The term V0 can be computed by

V0 = L2x1Tx1 + γ1x1Tx2∥x1∥1/2

+ L∥x1∥x2Tx2 +14 ∥x2∥

4

ð43Þ

(ii) CTSMC [24]: the gain adaptive scheme _L is definedas

_L = 0 ð44Þ

The parameters of the three methods are listed in Table 2.Besides, to validate the effectiveness of the improved algo-rithm, the method named ECATSMC whose gain adaptivelaw is defined as [28]:

_L = ηL sign ξð Þ −M L − L+½ �+ +M μ − L½ �+, ð45Þ

is also simulated, and its control parameters are the samewith those of IECATSMC.

Figure 2 shows the curves of the estimation error of zΘand the adaptive gains of the observer kΓ = ½kΓ1 kΓ2 �

T . FromFigure 2(a), we can see that the estimation error can converge

to the origin in a short time. Besides, it is shown inFigure 2(b) that the adaptive gains of the observer keepincreasing when the estimation error is large; then, theyremain constant.

Figure 3 denotes the tracking effect of yaw and pitchcommands of the four algorithms. As we can see fromFigure 3(a), they both can track the reference commands pre-cisely. After the control begins, CTSMC responses fastest intracking the commands, IEATSMC takes the second place,and ECATSMC takes the third place, ATSMC responds a lit-tle slower than the other three methods. What is more, wecan also see from Figure 3(a) that IECATSMC andECATSMC perform better in terms of tracking accuracy.CTSMC performs a little worse. The tracking effects for pitchcommand are shown in Figure 3(b). As we see in the picture,ECATSMC responds a little slower than the other threemethods. Despite this, ECATSMC can track the commandmore precisely than CTSMC and ATSMC. Besides, we canalso observe that the curve of pith angle under ATSMC devi-ates from the reference command curve around t = 8 s. Thisis due to the fact that the intensive disturbance is imposedaround that time. On the contrary, the tracking effects ofthe other three methods do not degrade.

Figure 4 shows the curves of the adaptive gains. We cansee from the picture that the gains of IECATSMC andECATSMC are slightly larger than 2 ∣ bΔ ∣ in almost thewhole control process. It should be noted that this approxi-mation can be easily adjusted by choosing the value of ε.We can also see that ATSMC can increase the gain to satisfy

_L =

ϖ1/ffiffiffiffiffiffiffi2γ1

p1/γ1ð Þ − 2Lx1Tx1+∥x1∥x2Tx2ð Þ/ L∗ − Lð Þ3

� � sign V0 − μð Þ, if L ≥ Lmin,χ, if L > Lmin:

8>>>>>>>:ð42Þ

9International Journal of Aerospace Engineering

L > 2∥bΔ∥ when 2∥bΔ∥ is larger than L. However, in othercases, the gain does not change. Besides, Figure 4 shows thatthe control gain L under CTSMC keep constant, although itsatisfies L > 2∥bΔ∥, its value is difficult to be set in practicalapplications given the upper boundary of disturbance isunknown.

Figure 5 shows the comparison between IECATSMCand ECATSMC and the curve of the exponential term.From Figure 5(a), we can see that at the beginning ofthe control process, the exponential term automativelyincreases to a large value to accelerate the changing rate

of the gain L. Actually, when there is a sudden changeoccurring in ∥bΔ∥, the exponential term can always adjustits value in time. As a result, IECATSMC reponses fasterthan ECATSMC, which is demonstrated in Figure 5(b).What is more, when the disturbance changes gently, theexponential term can adjust its value around 1. The resultis that IECATSMC has a smaller amplitude of oscillationthan ECATSMC, which enables the former one to sel-domly violate the condition of L > 2∥bΔ∥.

Figure 6 denotes the curves of equivalent canard anglecommands and equivalent canard angles. Because the

4 6 8 10 18161412Time (sec)

–100

0

100

δ yc (

deg)

20

ATSMC

IECATSMC

CTSMC

(a)

δ zc (

deg)

0 5 10 15

Time (sec)

–100

0

100

ATSMC

IECATSMC

CTSMC

(b)

δ y (d

eg)

0 5 10 15Time (sec)

–100

0

100

5.6 5.62 5.64

0.5

1

ATSMC

IECATSMC

CTSMC

(c)

δ z (d

eg)

0 5 10 15Time (sec)

–100

–50

0

50

13.24 13.26 13.28 13.3 13.32–0.6–0.4–0.2

00.20.4

ATSMC

IECATSMC

CTSMC

(d)

Figure 6: Equivalent canard angle commands.

10 International Journal of Aerospace Engineering

results of IECATSMC and ECATSMC are very similar,here, we only show the results of the former one for aclear comparison with the other two methods. Becausethe values of equivalent canard angle commands areclosely related with those of the control gains, we cansee from Figures 6(a) and 6(b) that IECATSMC greatlyattenuates the amplitude of chattering. This is alsoreflected in Figures 6(c) and 6(d).

Figure 7 shows how the gain is adjusted in the threephases in a clearer manner. As we see from the picture, thegains r1 = L and r2 = 0:5L continue to increase after controlbegins, and L > 2∥bΔ∥ is satisfied after t > t1. The gains reachtheir maximum on t = t2. After that, the equivalent controlinput ∥ueq∥ equals the disturbance; thus, the gains keepdecreasing until their curves follow the change of 2∥bΔ∥.Therefore, Figure 6 verifies the proof.

5.2. Monte Carlo Experiments. In order to further validate therobustness of the proposed control scheme, Monte Carloexperiments are performed 100 times. The external andinternal disturbances used in this subsection are the sameas that in Section 5.1, and the standard derivations of theerrors of the aerodynamic parameters are set to be 30% ofthe nominal values.

It can be observed from Figure 8 that the correspondingcurves are kept within small envelops, and all of them cantrack the desired commands accurately.

6. Conclusion

In this paper, a robust attitude tracking control scheme isproposed for the gliding-guided projectile. The proposedcontrol method does not require any information on the

0 5 10 15Time (sec)

4

6

8

10

12

𝛼 (d

eg)

0.3 0.35 0.410

10.5

11

𝛼c

(a)

0 5

𝛽c

10 15Time (sec)

–5

0

5

𝛽 (d

eg) –2 0.2 0.4 0.6

024

(b)

Figure 8: Tracking effect of yaw and pitch commands.

0 0.2 0.4 0.6 0.8 1 1.2Time (sec)

0

10

20

30

40

50

60Phase

1Phase

2Phase

3

t1 t2

r1 – IECATSMCr2 – IECATSMC ||△∣∣

||ueq∣∣

⌃

Figure 7: Gains in three phases.

11International Journal of Aerospace Engineering

bounds of disturbances. Simulation results show that the pro-posed method is not only more efficient in chattering attenu-ation than existing algorithms but also more precise intracking performance. Besides, in order to make the pro-posed attitude tracking controller be more robust, actuator-fault situation should be taken into consideration. Therefore,in our future work, we will extend the proposed controlscheme with a fault-tolerant approach.

Appendix

A. Proof of Lemma 3

Consider the following Lyapunov function candidate [33]:

V0 x1, x2ð Þ = L2x1Tx1 + γ1x1Tx2∥x1∥1/2

+ L∥x1∥x2Tx2 +14 ∥x2∥

4,

ðA:1Þ

and the above function can be rewritten as

V0 x1, x2ð Þ = ∥x1∥zTΓz +14 ∥x2∥

4, ðA:2Þ

where z = ½x1/∥x1∥1/2 x2�T , Γ =L2 γ1/2γ1/2 L

" #. To make Γ

be positive, γ1 is chosen to satisfy γ1 < 2L3/2. By utilizingthe relationship λminfΓg∥z∥2 ≤ zTΓz ≤ λmaxfΓg∥z∥2, we canhave

V0 x1, x2ð Þ = λmax Γf g∥z∥2 +14 ∥x2∥

4 = λmax Γf g ∥x1∥2+∥x1∥∥x2∥2� �

+ 14 ∥x2∥4 ≤

32 λmax Γf g∥x1∥

2 + 12 λmaxΓ +12

� �∥x2∥4

= ρTPρ,ðA:3Þ

where ρ = ½∥x1∥ ∥x2∥2�T , and the matrix

P =

32 λmax Γf g 0

0 12 λmax Γf g +12

� �2664

3775: ðA:4Þ

Then, (A.3) can be rewritten as

V0 x1, x2ð Þ ≤ λmax Pf g ∥x1∥2+∥x1∥∥x2∥2� �

+ 14 ∥x2∥4

≤ λmax Pf g ∥x1∥12+∥x2∥

� 4,

ðA:5Þ

On the other hand,

V0 x1, x2ð Þ ≥ λmin Pf g ∥x1∥2+∥x1∥∥x2∥2� �

+ 14 ∥x2∥4

≥ λmin Γf g∥x1∥2 +14 ∥x2∥

4,ðA:6Þ

Next, taking the derivative of V0ðx1, x2Þ, and substituting(27) into it, yields

_V0 x1, x2ð Þ = 2L2x1T +32 γ1∥x1∥

1/2x2 + Lx1T∥x1∥

−�

x2Tx2�_x1

+ γ1∥x1∥1/2x1T + 2L∥x1∥x2T�

+∥x2∥2x2T�_x2

= −γ1 L∥x1∥3/2−∥x1∥1/2x1T f + 0:5L�

− ∥x1∥1/2∥x2∥−1x1Tx2�− L∥x1∥ L∥x2∥−ð 2x2T f

�+ 32 γ1∥x1∥

1/2∥x2∥2 − 0:5L∥x2∥−ð x2T f�∥x2∥2

≤ −γ1 L−∥f∥+0:5Lð Þ∥x1∥3/2 − L L − 2∥f∥ð Þ∥x1∥∥x2∥+ 32 γ1∥x1∥

1/2 − ∥x2∥2 − 0:5L∥x2∥−∥f∥ð Þ∥x2∥2:ðA:7Þ

Recall that L > 2D0 and using the relationship ∥f∥≤D0, wecan obtain

_V0 ∥x1∥,∥x2∥ð Þ ≤ −∥x2∥ L L − 2D0ð Þð ∥x1∥−32 γ1∥x1∥

1/2∥x2∥�

+ 12 L −D0�∥x2∥2

� �− γ1

12 L −D0� �

∥x1∥3/2

= −γ112 L −D0� �

∥x1∥32−∥x2∥φTΨφ,

ðA:8Þ

where φ = ½jx1j1/2 ∣ x2 ∣ �T, and the matrix Φ is

Ψ =2L 12 L −D0� �

−34 L

−34 L

12 L −D0

26643775, ðA:9Þ

If L > 2D0, 0 < γ1 < ð4ffiffiffi2

p Þ/3 ffiffiffiLp ð0:5L −D0Þ holds, thenthe matrix Ψ is positive definite, which implies _V0ðx1, x2Þ isnegative definite, and the following inequality holds

λmax Ψð Þ ∥x1∥+∥x2∥2� �

≤ φTΨφ ≤ λmax Ψð Þ ∥x1∥+∥x2∥2� �

:

ðA:10Þ

By using the following well-known inequality,

xj jp + yj jp� �1/q ≤ 2 1/q−1/pð Þ xj jp + yj jp� �1/p, ðA:11Þ

12 International Journal of Aerospace Engineering

(A.8) can be rewritten as

_V0 ∥x1∥,∥x2∥ð Þ ≤ −γ112 L −Dp0� �

∥x1∥3/2−∥x2∥−λmin Ψð Þ ∥x1∥+∥x2∥2� �

≤ −γ112 L −Dp0� �

∥x1∥3/2−∥x2∥3−λmin Ψð Þ

≤ −E ∥x1∥3/2+∥x2∥3� �

≤ −E

22/3 ∥x1∥1/2+∥x2∥

� �3≤ −

E

22/3λ3/4max Pð ÞV

340 ∥x1∥,∥x2∥ð Þ,

ðA:12Þ

where E =min fγ1ðð1/2ÞL −D0Þ, λminðΨÞg. Let us denote theupper bound of λmaxðPÞ as �λmaxðPÞ, the lower bound of E asE , then (A.10) can be modified as

_V0 x1,x2ð Þ ≤ −E

22/3λ3/4max Pð ÞV3/40 x1,x2ð Þ

≤ −E

22/3�λ3/4max Pð ÞV3/40 x1, x2ð Þ:

ðA:13Þ

Therefore, the states x1 and x2 will converge to origin infinite time at most t2 ≤ t0 + t1 + ð16�λ3/4maxðPÞV1/40 ðt1ÞÞ/�E.

B. Proof of Theorem 7

The proof can be divided into two steps. Firstly, it will bedemonstrated that the closed-loop system (26) is boundedwith the existence of estimation error e. Next, the trajectoriesof the closed loop (26) without estimation error will be ana-lyzed. Since the uncertainty bΔ is bounded due to the bound-edness of _e, there exists a positive constant Ĉ, which satisfies∥bΔ∥≤C < Ĉ.

Choose a Lyapunov function candidate as V̂0ðeΘ, zΘÞ =V0ðeΘ, ẑΘÞ = L2eΘ2 + γ1jeΘj3/2ẑΘsign ðeΘÞ + L ∣ eΘ ∣ z∧Θ2 + ð1/4Þz∧Θ4. According to the anal-ysis in phases 1 and 2 in Theorem 4, the gain LðtÞ keepsincreasing until LðtÞ > 2Ĉ, and the flowing inequality canhold

_̂V0 eΘ, ẑΘð Þ ≤ −E

22/3λ3/4max Pð ÞV3/40 x1, x2ð Þ +Λ, ðB:1Þ

where

Λ = −2L2eΘTe −32 γ1∥

eΘ∥12z∧ΘTe

− Lz∧ΘT ẑΘeΘT∥eΘ∥

e: ðB:2Þ

Note that the estimation error e is bounded, assume∥e∥≤H, where H is a positive constant. Then, we obtain

Λ ≤HΞTRΞ ≤Hλmax Rð Þ ∥eΘ∥+∥ẑΘ∥2� �

, ðB:3Þ

where Ξ = ½∥eΘ∥1/2,∥z∧Θ∥�T , and the matrix R =

2L2 ð1/2Þγ1ð1/3Þγ1 L

" #, which is positive definite according

to the value of γ1. From (A.5), we can verify that V̂0 ≥ λminðΓÞ − ∥eΘ∥2 + 1/4∥ẑΘ∥4 ≥ λð∥eΘ∥2 + 1/4∥ẑΘ∥4Þ where λ =min fλminðΓÞ, 1/4g. Therefore,

Λ ≤Hλmax Rð Þ ∥eΘ∥+∥ẑΘ∥2� �

, ≤Hλmax Rð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2∥eΘ∥2+∥ẑΘ∥4

q,

≤ffiffiffi2

pHλmax Rð Þffiffiffi

λp V1/20 :

ðB:4Þ

Using (B.1) along with the above inequality, one has

_̂V0 ≤ −E

4λ3/4max Pð ÞV̂

3/40 +

ffiffiffi2

pHλmax Rð Þffiffiffi

λp V̂1/20 ,

= − E4λ3/4max Pð Þ

V1/20 V̂1/40 −

4ffiffiffi2

pHλmax Rð Þλmax Pð Þ3/4ffiffiffi

λp

E

!:

ðB:5Þ

Therefore, it is easy to verify that V̂0 will be con-

fined in a region which is a little larger than CV = fV̂0≤ ð4 ffiffiffi2p HλmaxðRÞÞ4λmaxðPÞ3Þ/λ2E4g. Thus, V̂0 is bounded,which means the states eΘ and ẑΘ are bounded. In Section3, we have shown that the estimation error e for the statezΘ will converge to zero in finite time. Without loss of gener-ality, the convergence time is set as To. So, ẑΘ = zΘ holds,when t > To. Thus, the closed-loop system (26) is reduced to

_eΘ = zΘ,

_zΘ = −r1eΘ∥eΘ∥

− r2ẑΘ∥ẑΘ∥

+ Δ,

0B@ ðB:6Þwhich is identical to (27). By following the analysis in Theo-rem 4, we can conclude that the attitude tracking error eΘ willconverge to zero at most tz ≤ To + t1 + ð16�λ3/4maxðPÞV1/40 ðt1ÞÞ/�E. What is more, the minimum possible value of the gain Lcan be kept in phase 3, which attenuates the amplitude ofchattering.

Nomenclature

α, β: Angle of attack, side slip angleθ, φ: Inclination angle, and deflection angle of flight

trajectoryϑ, ϕ: Pitch angle and yaw angleδz , δy: Deflection angle of canardsm: Mass of the projectilev: Speed of the projectileQ: Dynamic pressureD: Projectile diameter

13International Journal of Aerospace Engineering

E, C: Equatorial damping coefficient and polar dampingcoefficient [10]

l, s: Characteristic length and characteristic areamz′: The derivation of the static momentmzz′ : The derivation of equatorial damping momentmσ′: The derivation of control moment generated by

canardCαL, C

δL: Derivations of the lift induced by projectile body

and canardmμ, mμ′′: Magnus force and the derivation of Magnus

moment.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

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15International Journal of Aerospace Engineering

Gliding-Guided Projectile Attitude Tracking Controller Design Based on Improved Adaptive Twisting Sliding Mode Algorithm1. Introduction2. Gliding-Guided Projectile Attitude Kinematics Model3. Finite-Time Adaptive Observer for zΘ4. Main Results4.1. Improved Adaptive Twisting Sliding Mode Control Algorithm Design4.2. Stability Analysis for the Closed-Loop System

5. Simulation and Results5.1. Comparison5.2. Monte Carlo Experiments

6. ConclusionAppendixA. Proof of Lemma 3B. Proof of Theorem 7NomenclatureData AvailabilityConflicts of Interest